Abstract
We obtain new convolutions for quadratic-phase Fourier integral operators (which include, as subcases, e.g., the fractional Fourier transform and the linear canonical transform). The structure of these convolutions is based on properties of the mentioned integral operators and takes profit of weight-functions associated with some amplitude and Gaussian functions. Therefore, the fundamental properties of that quadratic-phase Fourier integral operators are also studied (including a Riemann–Lebesgue type lemma, invertibility results, a Plancherel type theorem and a Parseval type identity). As applications, we obtain new Young type inequalities, the asymptotic behaviour of some oscillatory integrals, and the solvability of convolution integral equations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Almeida, L.B.: Product and convolution theorems for the fractional Fourier transform. IEEE Signal Process. Lett. 4(1), 15–17 (1997)
Anh, P.K., Castro, L.P., Thao, P.T., Tuan, N.M.: Two new convolutions for the fractional Fourier transform. Wirel. Pers. Commun. 92(2), 623–637 (2017)
Anh, P.K., Tuan, N.M., Tuan, P.D.: The finite Hartley new convolutions and solvability of the integral equations with Toeplitz plus Hankel kernels. J. Math. Anal. Appl. 397(2), 537–549 (2013)
Beckner, W.: Inequalities in Fourier analysis. Ann. Math. 102, 159–182 (1975)
Brascamp, H.J., Lieb, E.H.: Best constants in Young’s inequality, its converse, and its generalization to more than three functions. Adv. Math. 20, 151–173 (1976)
Castro, L.P.: Regularity of convolution type operators with PC symbols in Bessel potential spaces over two finite intervals. Math. Nachr. 261–262, 23–36 (2003)
Castro, L.P., Duduchava, R., Speck, F.-O.: Finite interval convolution operators with transmission property. Integral Equ. Oper. Theory 52(2), 165–179 (2005)
Castro, L.P., Haque, M.R., Murshed, M.M., Saitoh, S., Tuan, N.M.: Quadratic Fourier transforms. Ann. Funct. Anal. AFA 5(1), 10–23 (2014)
Castro, L.P., Kapanadze, D.: Dirichlet-Neumann-impedance boundary value problems arising in rectangular wedge diffraction problems. Proc. Am. Math. Soc. 136(6), 2113–2123 (2008)
Castro, L.P., Kapanadze, D.: Wave diffraction by a half-plane with an obstacle perpendicular to the boundary. J. Differ. Equ. 254(2), 493–510 (2013)
Castro, L.P., Speck, F.-O.: Inversion of matrix convolution type operators with symmetry. Port. Math. (N.S.) 62(2), 193–216 (2005)
Castro, L.P., Saitoh, S.: New convolutions and norm inequalities. Math. Inequal. Appl. 15(3), 707–716 (2012)
Castro, L.P., Saitoh, S., Tuan, N.M.: Convolutions, integral transforms and integral equations by means of the theory of reproducing kernels. Opusc. Math. 32(4), 633–646 (2012)
Jain, P., Jain, S.: On Young type inequalities for generalized convolutions. Proc. A. Razmadze Math. Inst. 164, 45–61 (2014)
Jain, P., Jain, S.: Generalized convolution inequalities and application. Mediterr. J. Math. (2017). https://doi.org/10.1007/s00009-017-0961-3
Jain, P., Jain, S., Kumar, R.: On fractional convolutions and distributions. Integral Transforms Spec. Funct. 26(10), 885–899 (2015)
Nhan, N.D.V., Duc, D.T.: Weighted \(L^p\)-norm inequalities in convolutions and their applications. J. Math. Inequal. 2, 45–55 (2008)
Oinarov, R.: Boundedness and compactness of a class of convolution integral operators of fractional integration type. Proc. Steklov Inst. Math. 293, 255–271 (2016). [translation from Tr. Mat. Inst. Steklova 293, 263–279 (2016)]
Phong, D.H., Stein, E.M.: Models of degenerate Fourier integral operators and Radon transforms. Ann. Math. 140, 703–722 (1994)
Phong, D.H., Stein, E.M.: Oscillatory integrals with polynomial phases. Invent. Math. 110, 39–62 (1992)
Rudin, W.: Functional analysis, 2nd edn. McGraw-Hill, New York (1991)
Singh, A.K., Saxena, R.: On convolution and product theorems for FRFT. Wirel. Pers. Commun. 65(1), 189–201 (2012)
Stein, E.M.: Harmonic analysis. Princeton University Press, Princeton (1993)
Titchmarsh, E.C.: Introduction to the theory of Fourier integrals, 3rd edn. Chelsea Publishing Co., New York (1986)
Tuan, N.M., Huyen, N.T.T.: Applications of generalized convolutions associated with the Fourier and Hartley transforms. J. Integral Equ. Appl. 24(1), 111–130 (2012)
Vinogradov, O.L.: Sharp inequalities for approximations of convolution classes on the real line as the limit case of inequalities for periodic convolutions. Sib. Math. J. 58(2), 190–204 (2017) [translation from Sib. Mat. Zh. 58(2), 251-269 (2017)]
Wright, J.: From oscillatory integrals to complete exponential sums. Math. Res. Lett. 18(2), 231–250 (2011)
Zayed, A.I.: A convolution and product theorem for the fractional Fourier transform. IEEE Signal Process. Lett. 5(4), 102–103 (1998)
Acknowledgements
The first-named author was supported in part by FCT–Portuguese Foundation for Science and Technology through the Center for Research and Development in Mathematics and Applications (CIDMA) at Universidade de Aveiro, within the Project UID/MAT/04106/2013. The two last named-authors were partially supported by the Viet Nam National Foundation for Science and Technology Development (NAFOSTED).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Castro, L.P., Minh, L.T. & Tuan, N.M. New Convolutions for Quadratic-Phase Fourier Integral Operators and their Applications. Mediterr. J. Math. 15, 13 (2018). https://doi.org/10.1007/s00009-017-1063-y
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-017-1063-y
Keywords
- Convolution
- Young inequality
- oscillatory integral
- convolution integral equation
- fractional Fourier transform
- linear canonical transform